Lectures on Nuclear and Hadron Physics - UM · Lectures on Nuclear and Hadron Physics Dispersion...

Lectures on Nuclear and Hadron Physics


J. A. Oller

Departamento de FısicaUniversidad de Murcia

July 27, 2011


Contents

Outline Lecture I

1 Introduction

2 Dispersion Relations

3 S-matrix

4 Scalar Form Factor of the Pion

5 γγ → π0π0


Contents

Introduction

In the chiral limit mu = md = ms = 0 the QCD Lagrangian isinvariant under UL(3) ⊗ UR(3) symmetry in the quark flavours ortypes.

SUL(3) ⊗ SUR(3) → SUV (3) is Spontaneously Broken.

Goldstone bosons appear π, K , ηMass gap:Mπ ∼ 140 MeV ≪ Mρ ∼ 770 MeVMK ∼ 496 MeVMη ∼ 550 MeV

Conservation of axial-vector current: ∂µ (qγµγ5λaq) = 0 ,

〈0|qγµγ5λaq|πb(p)〉 = i δabFπ pµ

∂µ〈0|qγµγ5λaq|πb(p)〉 = i δabFπ p2 = 0


Contents

p2 = 0 → M2b = 0

L + R corresponds to qγµqUV (1) ≡ UL+R Conserved Baryon Number.

L − R corresponds to qγµγ5qUA(1) ≡ UL−R Neither Conserved nor Goldstone Boson.

Puzzle: (if UA(1) were spontaneously broken)Goldstone mode: There should be an η0 with a mass <

√3mπ

but η is much heavier Weinberg PRD’75


Contents

QCD Anomaly

The ninth axial singlet current has an anomalous divergence Adler

PR’69 Fujikawa PRD’80

Jµ (0)5 = qγµγ5q

∂µJµ (0)5 =

g2

16π2

1

Nc

Trc(GµνGµν)

Large Nc QCD ’t Hooft NPB’74, Witten NPB’79 Nc → ∞,g2Nc → constant

UL(NF ) ⊗ UR(NF ) → UL+R(NF ) Coleman, Witten PRL’80

Entire Nonet of Goldstone bosons results.


Dispersion Relations


Causality conditions → Dispersion Relations

Analytical properties of transient amplitudes

H. A. Kramers, Atti Cong. Intern. Fisica, Como, 2 (1927) 545;R. Kronig, Opt. Soc. Amer. 12 (1926) 547.Scattering of light by a dispersive medium.The name ‘dispersion relation’ derives from this first application.CAUSAL STATEMENT: Polarization P(t)

P(t) = 0 for t < 0

Fourier Transform E (ν) = n(ν) − 1:

E (ν) = 4π

∫ +∞

−∞

dtP(t)e iνt .



Dispersion Relation:

ℜE (ν) =1

πP

∫ +∞

−∞

dν ′ℑE (ν ′)

ν ′ − ν.

Proof: Essential point

ν = ν1 + i ν2, ν2 > 0 ,

E (ν) = 4π

∫ +∞

−∞

dtP(t) exp[iν1t − ν2t] .

It converges because P(−∞) = 0.• Cauchy’s theorem: (Assume E (ν) → 0 for ν → ∞)

E (ν + iǫ) =1

2πi

∫

C

dν ′ E (ν ′)

ν ′ − ν − iǫ(1)

C

ν + iǫ



0 =1

2πi

∫

C

dν ′ E (ν ′)

ν ′ − ν + iǫ

Complex

Conjugate

→ 0 =1

2πi

∫

C

dν ′ E ∗(ν ′)

ν ′ − ν − iǫ(2)

• Subtracting Eqs.(1), (2):

E (ν) = limǫ→0+

1

π

∫ +∞

−∞

dν ′ ℑE (ν ′)

ν ′ − ν − iǫ(3)

Adding Eqs.(1), (2):

E (ν) = limǫ→0+

−i

π

∫ +∞

−∞

dν ′ ℜE (ν ′)

ν ′ − ν − iǫ(4)

Taking into account

∫ +∞

−∞

dxg(x)

x − x0 ± iǫ= P

∫ +∞

−∞

dxg(x)

x − x0∓ iπg(x0)



From Eqs. (3) and (4), respectively:

ℜE (ν) = P1

π

∫ +∞

−∞

dν ′ℑE (ν ′)

ν ′ − ν, (5)

ℑE (ν) = P1

π

∫ +∞

−∞

dν ′ℜE (ν ′)

ν ′ − ν. (6)

Eq. (5) is by far much more useful.It expresses the Dispersive part (ℜE (ν)) in terms of the Absorptivepart (ℑE (ν)), once it is known for all frequencies.

The same results are obtained directly from

E (ν + iǫ) =1

2πi

∫

C

dν ′ E (ν ′)

ν ′ − ν − iǫ

Applying

1

x − x0 − iǫ→ P

∫dx

x − x0+ iπδ(x − x0) .



Subtractions

Let’s relax the assumption E (ν) → 0 for ν → ∞• Now:

limν→∞

E (ν)

ν= 0 (7)

One Subtraction

E (ν + iǫ) − E (ν0 + iǫ) =1

2πi

∫

C

dν ′

[E (ν ′)

ν ′ − ν − iǫ− E (ν ′)

ν ′ − ν0 − iǫ

]

=ν − ν0

2πi

∫

C

dν ′ E (ν ′)

(ν ′ − ν − iǫ)(ν ′ − ν0 − iǫ)

In the same way as before:

ℜE (ν) −ℜE (ν0) = P1

π

∫ +∞

−∞

dν ′

[ℑE (ν ′)

ν ′ − ν− ℑE (ν ′)

ν ′ − ν0

]

=ν − ν0

πP

∫ +∞

−∞

dν ′ ℑE (ν ′)

(ν ′ − ν)(ν ′ − ν0)



The addition of further subtractions is iterative. Assume:

limν→∞

E (ν)

ν2= 0 .

Two subtractions: One considers

E (ν) − E (ν0) −ν − ν0

ν1 − ν0E (ν1) − E (ν0)

At ν = ν0 → E (ν0). At ν = ν1 → E (ν1).

E (ν) − E (ν0) −ν − ν0

ν1 − ν0E (ν1) − E (ν0)

=1

2πi

∫

C

dν ′E (ν ′)

[1

ν ′ − ν− 1

ν ′ − ν0− ν − ν0

ν1 − ν0

(1

ν ′ − ν1− 1

ν ′ − ν0

)]

=(ν − ν0)(ν − ν1)

2πi

∫

C

dν ′ E (ν ′)

(ν ′ − ν)(ν ′ − ν0)(ν ′ − ν1).



ℜE (ν) −ℜE (ν0) −ν − ν0

ν1 − ν0ℜE (ν1) −ℜE (ν0)

=(ν − ν0)(ν − ν1)

πP

∫

C

dν ′ ℑE (ν ′)

(ν ′ − ν)(ν ′ − ν0)(ν ′ − ν1).



Dispersion relations were applied to particle physics following asuggestion by R. Kronig, Physica 12 (1946) 543:

Causality requirements should be added to the usual conditions onthe S-matrix, namely, Lorentz invariance and unitarityM. Gell-Mann, M. L. Goldberger and W. E. Thirring, Phys. Rev.95 (1954) 1612The causality requirement: “the commutator of twoHeisenberg operators for the field in question shall vanish ifthe operators are taken at space-like points.”

[φ(x1), φ(x2)] = 0 for (x1 − x2) space-like

They demonstrated that the amplitude for the forward scatteringof photons by nucleons satisfies:

ℜA(ν) =1

π

∫ +∞

−∞

dν ′ℑA(ν ′)

ν ′ − ν


S-matrix

Sba = 〈Ψ−b |Ψ+

a 〉 = limt1 → −∞t2 → +∞

〈φb|U(t2, t1)|φa 〉

S-matrix is a Unitary Operator

S · S† = S†S = I

T -matrix: S = 1 + iT

Unitarity: S† · S = 1

〈φb|T |φa 〉 − 〈φb|T †|φa 〉 = i∑

c

∫dQc〈φb|T †|φc 〉〈φc |T |φa 〉 .

In the sum only those states which are open contribute!Phase space

dQα = Πnα

i=1

∫ [d3pi

2p0i (2π)3

](2π)4δ(Pb − Pa) ≡ dα(2π)4δ(Pα − Pa)


S-matrix

Forward scattering

Very useful case: a = b

ℑTaa =1

2

∑

c

∫dQc |〈φc |T |φa 〉|2

It provides the imaginary part for positive energy above threshold

Unitarity: S · S† = 1

ℑTaa =1

2

∑

c

∫dQc |〈φa|T |φc 〉|2

Important result:

∑

c

∫dQc |〈φc |T |φa 〉|2 =

∑

c

∫dQc |〈φa|T |φc 〉|2


S-matrix

Rate of all reactions produced by an initial state α

Γα =1

cα

∑

β

∫dQβ |Tβα|2

From unitarity we then obtain

∑

β

∫dβcα

dΓ(α → β)

dβ=

∑

β

∫dβcβ

dΓ(β → α)

dβ

Pαdα Probability to find the system within volume dα

dPα

dt=

∑

β

∫dβPβ

dΓ(β → α)

dα− Pα

∑

β

∫dβ

Γ(α → β)

dβ


S-matrix

Boltzmann H-theorem

Entropy: −∫

dαPα ln(Pα/cα)It follows also from forward scattering unitarityS. Weinberg, The Quantum Field Theory of Fields. Vol. II:

− d

dt

∫dαPα ln(Pα/cα) ≥ 0


S-matrix

Partial waves

Unitarity is most simply expressed in terms of Partial WavesLacour, Meißner, J.A.O, Ann. Phys. 326 (2011) 241

|p, σ1σ2 〉 =√

4π∑

J,ℓ,S ,m

(σ1σ2s3|s1s2S)(ms3µ|ℓSJ)Y mℓ (p)∗|J µ ℓS s1 s2 〉

Two-particle scattering

a(p1) + b(p2) → c(p3) + d(p4)

Spinless case

Tℓ(s) =1

2

∫ +1

−1d cos θ Pℓ(cos θ)T (s, cos θ)

ℑTℓ(s)fi =∑

a

Tℓ(s)iaTℓ(s)∗fa

|p|a8π

√s


S-matrix

Crossing

With unitarity alone we do not posses enough information for thestandard dispersion relation (ν ′ > 0)

ℜA(ν) =1

π

∫ +∞

−∞

dν ′ℑA(ν ′)

ν ′ − ν

since ν ′ < 0 is also required.Relativistic quantum field theory → Crossing Symmetry.Another condition that together with unitarity and dispersionrelations allows dynamical calculations

A B

C D A + B → C + D s-channel

A + C → B + D t-channel

A + D → B + C u-channel

s = (pA + pB)2 , t = (pA − pC )2 ,u = (pA − pD)2


S-matrix

Mandelstam Variables: Only two are independents + t + u = m2

A + m2B + m2

C + m2D

A(pA) + B(pB) → C (pC ) + D(pD) s-channel

A(pA) + C (−pC ) → B(−pB) + D(pD) t-channel

A(pA) + D(−pD) → B(−pB) + C (pC ) u-channel

Particle→Antiparticle and reverse four-momentumThe amplitudes for the three reactions are given by the sameanalytic functionPhysical processes correspond to certain values of s, t(disconnected physical regions)


S-matrix

sc → u

III

III


S-matrix

E.g.

Nπ0 → Nπ0

s- and u-channel processes are the sameFor forward scattering:

ℜA(−ν) = ℜA(ν)

ℑA(−ν) = −ℑA(ν)ℜA(ν) =

∫ +∞

−∞

dν ′ℑA(ν ′)

ν ′ − ν

We have the needed information to apply the dispersion relationν ′ ∈] −∞,−mπ] and [mπ, +∞]

Additionally one also has the one-nucleon pole

g2

2mN

1

ν − m2π/(2mN)


S-matrix

The application of crossing is not always so straightforward forobtaining the input in the dispersion relation.

Typical example is fixed t dispersion relations

t = −2q2(1 − cos θ)

q2 =(s ′ − (mN + mπ)2)(s ′ − (mN − mπ)2)

4s ′

As s ′ moves as integration variable cos θ will be less than −1 forq2 → 0 (near threshold)

The extension to unphysical values of cos θ can be done makinguse of the partial wave expansion.

However, its convergence its restricted to the Lehmann ellipseH. Lehmann, Nuovo Cim. Suppl. 14 (1959) 153


S-matrix

πN scattering0 < −t <

32

3

2mN + mπ

2mN − mπm2

π ∼ 32

3m2

π


S-matrix

The principle of Maximum Analyticity

S. Mandelstam, Rep. Prog. Phys. 25 (1962) 99“The scattering amplitude is analytic in all its variablesexcept at those points where singularities arise as aconsequence of the unitarity condition.” (including thecrossed channels.)This is fulfilled to all orders in perturbation theory.

Direct Crossed


Scalar Form Factor of the Pion


L. Roca and J.A.O., Phys. Lett. B 651 (2007) 139

Γπ(t) =

∫d4x e−i(q′−q)x〈π(q′)|

(muu(x)u(x) + md d(x)d(x)

)|π(q) 〉 , t

t = (q′ − q)2

Quadratic scalar radius of the pion 〈 r2 〉πs

Γπ(t) = Γπ(0)

1 +

1

6t〈 r2 〉πs + O(t2)

• 〈 r2 〉πs contributes 10% to the a00 and a2

0 scattering lengths fromRoy equations+CHPT to two loops (2% of precision). It is a bigcontribution.G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 (2001)125 (CGL)



• It gives ℓ4 that controls the departure of Fπ from its value in thechiral limit

〈 r2 〉πs =3

8π2F 2π

(ℓ4 −

13

12

)

Fπ = F

1 +

m2π

16π2F 2π

ℓ4

• Controversy between:The ‘canonical’ value: 〈 r2 〉πs = 0.61 ± 0.04 fm2

Donoghue, Gasser, Leutwyler, Nucl. Phys. B 343 (1990) 341Based on solving the two-channel Muskhelishvili-Omnes equations

F. Yndurain’s approach based on the Omnes representation ofΓπ(t) and high energy QCD behavior〈 r2 〉πs = 0.75 ± 0.07 fm2



• Change in the ππ scattering lengths:

δa00 = +0.027∆r2 δa2

0 = −0.004∆2r 〈 r2 〉πs = 0.61(1 + ∆2

r ) fm2

∆r2 = +0.23

δa00 = +0.006 δa2

0 = −0.001

CGL:

a00 = 0.220 ± 0.005 a2

0 = −0.0444 ± 0.0010 fm

precision 2%



scalar

source

Unitarity Cutreplacements

π

π π

π

Γπ(t) only has RHC or Unitarity cut∞ > t ≥ 4m2

π

2π 4π 6π κκ

t C



Dispersion relation:

∮

C

Γπ(z)

z − t= 0

2πiΓπ(t) =

∫ ∞

4m2π

ds

(Γπ(s + iǫ)

s + iǫ − t− Γπ(s − iǫ)

s − iǫ − t

)

Schwartz’s reflection principle:If f (z) is real along an interval of the real axis and is analytic then:f (z∗) = f (z)∗

Γπ(t) =1

π

∫ ∞

4m2π

dsℑΓπ(s)

s − t



One hardgluon; 1/t

Brodsky-Farrar QCDcounting rulesThe dispersion integral isconvergent



Omnes representationLet F (t) an analytic function except for the presence of the RHC

1 Remove the zeroes and poles in F (t) in the complex plane

g(t) =F (t)

P(t)

P(t) =F (0)

s1 · · · sn(s1 − t)(s2 − t) · · · (sn − t)

2 Perform a dispersion relation of

f (t) = logF (t)

P(t)

f (t + iǫ) − f (t − iǫ) = logF (t + iǫ)

P(t)− log

F (t − iǫ)

P(t)

= 2i argF (t + iǫ)

P(t)|F (t + iǫ)| = |F (t − iǫ)|



φ(t) = arg F (t)P(t)

φ(t) is continuous

φ(4m2π) = 0

logF (t)

P(t)= γ0 +

t

π

∫ ∞

4m2π

dsφ(s)

s − t − iǫ

F (t) = P(t) expt

π

∫ ∞

4m2π

dsφ(s)

s(s − t − iǫ)

With n the degree of P(t) then

F (t) → (−1)ne iφ(+∞) tn−φ(+∞)/π

φ(+∞) → (n + 1)π to guarantee that F (t) → 1/t (QCD)



Watson final state theorem

S = 1 + i2ρ(F + T )

SS† = 1

• Elastic case (only ππ intermediate state)• Keep only linear terms in F . There is only one source!

ℑΓπ(t) = Γπ(t)ρ(t)T ∗ππ(t)

Since the ℑΓπ(t) is real the phase of F (t) (φ(t)) and of Tππ(t)(δ(t)) are the same (modulo π)



Corolary

tππ = ρTππ = sin δπ e iδπ , δπ(4m2π) = 0, δπ is continuous and at

most differs by modulo π from the phase of Tππ

This happens when δπ crosses π (sin δπ < 0)

F (t) = P(t) exp1

π

∫ ∞

4m2π

dsφ(s)

s(s − t)

φ(s) = δ(s) s < 4m2K



Yndurain’s method: F. Yndurain, Phys. Lett. B 578 (2004) 99The form factor is assumed to be free of zeroes:

Γπ(t) = Γπ(0) exp

[1

π

∫ ∞

4m2π

dsφ(s)

s(s − t)

]

〈 r2 〉πs =6

π

∫ ∞

4m2π

dsφ(s)

s2

However, this result is very unstable and depends on the exactvalue of δπ(4m2

K ) whether is larger or smaller than π

Actual values of δπ(4m2K ) are very close to π



0 200 400 600 800 1000 1200

t1/2

(MeV)

0

1

2

3

4

5

6| Γ

(t)

/ Γ(0

) |

δ (sK

)<

δ (sK

)>π, P(t)=Γ (0)(s1- t)/s

1

δ (sK

)>π, P(t)=Γ (0)ref. [7] (δ (s

K)>π)π

ππππ

ππ

ππ



Continuity is restored by including a zero in the form factor at s1 ifδ(sK ) ≥ π

Γπ(t) = Γπ(0)s1 − t

s1exp

[1

π

∫ ∞

4m2π

dsφ(s)

s(s − t)

]

〈 r2 〉πs = − 6

s1+

6

π

∫ ∞

4m2π

φ(s)

s2ds

δ(s1) = π s1 < 4m2K



1000 1200 1400 1600 1800 2000

s1/2

(MeV)

0

50

100

150

200

250

300

350

400

450

500

ϕπ , ( δπ(sK)< 180

o )

δ(+) , ( δπ(sK)< 180

o )

ϕπ , ( δπ(sK)> 180

o )

δ(+) , ( δπ(sK)> 180

o )

δ(4m2K ) < π φ(∞) → π δ(4m2

K ) ≥ π φ(∞) → 2π



φas ≃ π

(n ± 2dm

log(s/Λ2)

)

dm = 12/(33 − 2nf ) ≃ 1/2Λ is the scale QCD parametern = 1, 2 for δπ(4m2

K ) < π, ≥ π

〈 r2 〉πs = 0.63 ± 0.05 fm2

Compatible with CGL 〈 r2 〉πs = 0.61 ± 0.04 fm2

Both methods are compatible


γγ → π0π

0

γγ → π0π0

L.Roca, J.A.O., Phys. Lett. B 659 (2008) 201

Similar problems appear in the literature for this reaction due tothe use of the Omnes function with its critical behavior dependingof δ(4m2

K ) > π or < π

250 300 350 400 450 500 550 600 650 700 750 800

s1/2

(MeV)

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

σ (n

b)

δ (sK

) > π, eq.(2.4)

δ (sK

) > π, eq.(2.2)

γγ π0 0π

ππ Pennington, Phys. Rev. Lett. 97

(2006) 011601

Morgan,Pennington, Phys. Lett. B 272

(1991) 134

Donoghue,Holstein, Phys. Rev. D 48

(1993) 137


γγ → π0π

0

The amplitude F (s) has RHC and LHC. The latter is given by L(s)

F (s) − L(s)

has only RHC

Ω(s) =

(1 − θ(s1)

s

s1

)exp

[s

π

∫ ∞

4m2π

φ0(s′)

s ′(s ′ − s)ds ′

]

Twice-subtracted dispersion relation for

(F (s) − L(s))/Ω(s)

F (s) = L(s) + c0sΩ(s) +s2

πΩ(s)

∫ ∞

4m2π

L0(s′) sin φ0(s

′)

s ′2(s ′ − s)|Ω(s ′)|ds ′

+ θ(s1 − 4m2K )

Ω0(s)

Ω0(s1)

s2

s21

(F0(s1) − L0(s1)) .


γγ → π0π

0

Low’s theorem allows to fix c0

F.E. Low, Phys. Rev. 96 (1954) 1428

The γγ → ππ amplitudes tend linearly to the Born term for s → 0

P

p 1

p 2k 2

k 1

P

γ

γ

P

P

P+

P+

P+

P+

P−

P−P−

P−

Other sources for L(s) are Axial vector (a1) and vector (ρ, ω)resonances


γγ → π0π

0

250 300 350 400 450 500 550 600 650 700 750 800

s1/2

(MeV)

0

4

8

12

16

20

σ (n

b)

γγ π0 0

ChPT at two loops

PY

CGL

ChPT at one loop

π

The strong interactions affect the outputCGL: Colangelo, Gasser, Leutwyler Nucl. Phys. B 603 (2001) 125PY: Pelaez, Yndurain, Phys. Rev. D 71 (2005) 074016

This reaction can be used to favor one or other strong amplitudefor meson-meson S-waveProjected precise experiments on γγ → π0π0, → π+π− for thenear future (BELLE, Frascatti)

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